Question

1.

i. Show that P is closed under intersection. (This means that if languages J and L are both in P, then so is J ∩ L.)

ii. Show that JL (the concatenation of J and L) is in P. (Here J and L are in P as in part (i), and JL = { xy | x ∈ J and y ∈ L }. ) Here you should calculate the complexity of your P-time algorithm for JL.

iii. Show that NP is closed under union

Answer 1

1.

iii)

Assume language X and language Y are in NP, we wanted to show X union Y is in NP

Because X and Y are in NP, there exists non-deterministic Turing machine X and non-deterministic Turing machine Y that verifies X and Y, respectively.

Rename the states in Y so that it does not have the same name as in X.

Build a new non-deterministic Turing machine U as follow:

Create a new start state, for each input symbol, it write the same symbol back to the tape, and non-deterministically go to the start state of X or the start state of Y. Take the union of state and transition function from both X and Y, that is a non-deterministic Turing machine U.

For any string in X, non-deterministic Turing machine U can take a non-deterministic jump to the start state of X, and follow the non-deterministic Turing machine X to finally get accepted.
For any string in Y, non-deterministic Turing machine U can take a non-deterministic jump to the start state of Y, and follow the non-deterministic Turing machine Y to finally get accepted.
For any string that is neither in X nor in Y, non-deterministic jump to either X or Y cannot lead to acceptance.

Therefore we showed the non-deterministic Turing machine U precisely accept X union Y.

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